Definition 39.11.1. Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a scheme over S, and let a : G \times _ S X \to X be an action of G on X.
We say X is a pseudo G-torsor or that X is formally principally homogeneous under G if the induced morphism of schemes G \times _ S X \to X \times _ S X, (g, x) \mapsto (a(g, x), x) is an isomorphism of schemes over S.
A pseudo G-torsor X is called trivial if there exists an G-equivariant isomorphism G \to X over S where G acts on G by left multiplication.
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